Conformal mapping of rectangular heptagons II

TL;DR

This paper introduces an analytical method for conformally mapping rectangular heptagons to a half-plane using Riemann theta functions, enabling applications in fluid flow and capacitor capacity calculations.

Contribution

It presents a novel approach leveraging abelian integrals and Riemann theta functions for conformal mapping of complex polygons, expanding analytical tools in geometric function theory.

Findings

Successfully maps rectangular heptagons to half-plane

Calculates 2D ideal fluid flow above rectangular surfaces

Determines capacities of multi-component rectangular condensers

Abstract

A new analytical method for the conformal mapping of a rectangular heptagon with a straight angle at infinity to a half plane and back is proposed. The method is based on the observation that SC integral in this case is an abelian integral on a genus two curve, so it may be represented in terms of Riemann theta functions. The approach is illustrated by the computation of 2D flow of ideal fluid above rectangular underlying surface and the computation of the capacities of multi component rectangular condensers with axial symmetry.